Multivariate limited translation empirical Bayes estimators

The paper develops multivariate limited translation empirical Bayes estimators of the normal mean vector which serve as a compromise between the empirical Bayes and the maximum likelihood estimators. These compromise estimators perform better than the regular empirical Bayes estimators, in a frequentist sense, when there is wide departure of an individual observation from the grand average.     

Bayes estimators of the average failure rates

In the present paper we have proposed a Bayesian approach for making inferences from accelerated life tests which do not require distributional assumptions     

Hierarchical Bayes estimation of mortality rates for disease mapping

Mapping of incidence rates or mortality rates (relative risks) from diseases like cancer and leukemia is of primary importance in an epidemiological study. The usual procedure is to map the standardized mortality ratio (SMR) across different geographical regions. Direct use of SMR may not be worthwhile, particularly for small places, as it does not take into account the high variability for different population sizes over different regions and the spatial patterns of the regions under study. In this paper a hierarchical Bayes approach is presented in smoothing the relative risks and providing the measures of uncertainty associated with these estimates of relative risks.     

Convergence rates of empirical Bayes estimation in exponential family

The convergence rates of empirical Bayes estimation in the exponential family are studied in this paper. We first develop an approach for obtaining the lower bound of empirical Bayes estimators. As an application of the approach, we demonstrate that O(n⁻¹) is the lower bound rate for priors with bounded compact support. Second, we construct an empirical Bayes estimator using kernel sequence method and show that it has a rate of convergence of $\text{O}\text{(n}{}^{\mathrm{-1}}\text{(}\text{ln}\text{n)}{}^8\text{)}$. This upper bound rate is much faster compared to the earlier results published in the literature under the same assumption.     

Bayes minimax ridge regression estimators

The problem of estimating of the vector β of the linear regression model y = Aβ + ? with ? ～ N_p(0, σ²I_p) under quadratic loss function is considered when common variance σ² is unknown. We first find a class of minimax estimators for this problem which extends a class given by Maruyama and Strawderman (2005 Maruyama, Y., and W. E. Strawderman. 2005. A new class of generalized Bayes minimax ridge regression estimators. Annals of Statistics 33:1753–70.[Crossref], [Web of Science ®] , [Google Scholar]) and using these estimators, we obtain a large class of (proper and generalized) Bayes minimax estimators and show that the result of Maruyama and Strawderman (2005 Maruyama, Y., and W. E. Strawderman. 2005. A new class of generalized Bayes minimax ridge regression estimators. Annals of Statistics 33:1753–70.[Crossref], [Web of Science ®] , [Google Scholar]) is a special case of our result. We also show that under certain conditions, these generalized Bayes minimax estimators have greater numerical stability (i.e., smaller condition number) than the least-squares estimator.     

Convergence rates for empirical bayes estimators of parameters in multi-parameter exponential families

This paper obtains the convergence rates of the empirical Bayes estimators of parameters in the multi-parameter exponential families. The rates can approximate to 0(n=¹) arbitrarily. The paper presents the multivariate orthogonal polynomials which are continuous on the total space R^p.     

Convergence rates for empirical bayes estimators of parameters in linear regressin models

In this paper, the convergence rates of the EB estimators of the regression coefficients and the error variance in a linear model are obtained. The rates can approximate to O(n¹) arbitrarily. The convergency of the EB estimators of the regression coefiicients and the variance components in a variance component model is also investigated. The investigation makes use of the results concerning the convergence rates of the EB estimators of the parameters in multi-parameter exponential families.     

The equivalence of Bayes and robust Bayes estimators for various loss functions

The problem of Bayes and robust Bayes estimation for various bounded and/or symmetric loss functions in a normal model with conjugate and non-informative prior distributions is considered. The prior distribution is not fully specified and covers the conjugate family of priors. It is of interest to know that the Bayes and robust Bayes estimators for symmetric losses are the same as those for the standard square-error loss function.     

Bayes and empirical Bayes shrinkage estimation of regression coefficients

For a moderate or large number of regression coefficients, shrinkage estimates towards an overall mean are obtained by Bayes and empirical Bayes methods. For a special case, the Bayes and empirical Bayes shrinking weights are shown to be asymptotically equivalent as the amount of shrinkage goes to zero. Based on comparisons between Bayes and empirical Bayes solutions, a modification of the empirical Bayes shrinking weights designed to guard against unreasonable overshrinking is suggested. A numerical example is given.     

Pseudo-Bayes and pseudo-empirical Bayes estimators in randomized response sampling

In this article, new pseudo-Bayes and pseudo-empirical Bayes estimators for estimating the proportion of a potentially sensitive attribute in a survey sampling have been introduced. The proposed estimators are compared with the recent estimator proposed by Odumade and Singh [Efficient use of two decks of cards in randomized response sampling, Comm. Statist. Theory Methods 38 (2009), pp. 439–446] and Warner [Randomized response: A survey technique for eliminating evasive answer bias, J. Amer. Statist. Assoc. 60 (1965), pp. 63–69].     

Adaptively leveraging external data with robust meta-analytical-predictive prior using empirical Bayes

The robust meta-analytical-predictive (rMAP) prior is a popular method to robustly leverage external data. However, a mixture coefficient would need to be pre-specified based on the anticipated level of prior-data conflict. This can be very challenging at the study design stage. We propose a novel empirical Bayes robust MAP (EB-rMAP) prior to address this practical need and adaptively leverage external/historical data. Built on Box's prior predictive p-value, the EB-rMAP prior framework balances between model parsimony and flexibility through a tuning parameter. The proposed framework can be applied to binomial, normal, and time-to-event endpoints. Implementation of the EB-rMAP prior is also computationally efficient. Simulation results demonstrate that the EB-rMAP prior is robust in the presence of prior-data conflict while preserving statistical power. The proposed EB-rMAP prior is then applied to a clinical dataset that comprises 10 oncology clinical trials, including the prospective study.     

Three strings of inequalities among six Bayes estimators

We discover three interesting strings of inequalities among six Bayes estimators, where for the parameter space (0, 1), (0, ∞), and ( ? ∞, ∞), each case has a string of inequalities. The three strings of inequalities only depend on the loss functions, and the inequalities are independent of the chosen models and the used priors provided the Bayes estimators exist. Therefore, they exist in a general setting which makes them quite interesting. Finally, the numerical simulations exemplify the two strings of inequalities defined on (0, 1) and (0, ∞), and that there does not exist a string of inequalities among the six smallest posterior expected losses.     

Empirical Bayes procedures for stabilizing maps of U.S. cancer mortality rates

"The geographic mapping of age-standardized, cause-specific death rates is a powerful tool for identifying possible etiologic factors, because the spatial distribution of mortality risks can be examined for correlations with the spatial distribution of disease-specific risk factors. This article presents a two-stage empirical Bayes procedure for calculating age-standardized cancer death rates, for use in mapping, which are adjusted for the stochasticity of rates in small area populations. Using the adjusted rates helps isolate and identify spatial patterns in the rates. The model is applied to sex-specific data on U.S. county cancer mortality in the white population for 15 cancer sites for three decades: 1950-1959, 1960-1969, and 1970-1979. Selected results are presented as maps of county death rates for white males."     

An empirical Bayes approach to outliers

A formulation of the problem of detecting outliers as an empirical Bayes problem is studied. In so doing we encounter a non-standard empirical Bayes problem for which the notion of average risk asymptotic optimality (a.r.a.o.) of procedures is defined. Some general theorems giving sufficient conditions for a.r.a.o. procedures are developed. These general results are then used in various formulations of the outlier problem for underlying normal distributions to give a.r.a.o. empirical Bayes procedures. Rates of convergence results are also given using the methods of Johns and Van Ryzin (1971, 1972).     

Flexible empirical Bayes estimation for wavelets

Wavelet shrinkage estimation is an increasingly popular method for signal denoising and compression. Although Bayes estimators can provide excellent mean-squared error (MSE) properties, the selection of an effective prior is a difficult task. To address this problem, we propose empirical Bayes (EB) prior selection methods for various error distributions including the normal and the heavier-tailed Student t -distributions. Under such EB prior distributions, we obtain threshold shrinkage estimators based on model selection, and multiple-shrinkage estimators based on model averaging. These EB estimators are seen to be computationally competitive with standard classical thresholding methods, and to be robust to outliers in both the data and wavelet domains. Simulated and real examples are used to illustrate the flexibility and improved MSE performance of these methods in a wide variety of settings.     

Bayes and admissibility properties of estimators in truncated parameter spaces

For the problem of estimating a parameter θ when θ is known to lie in a closed, convex subset D of R^k, conditions are given under which estimators δ of θ cannot be Bayes estimators, as well as conditions under which δ is inadmissible. The estimators considered are so-called “boundary estimators”. Maximum-likelihood estimators in truncated parameter spaces are examples to which our results often apply. For the special case when k = 1 and D is compact, two classes of estimators dominating the inadmissible ones are constructed. Some examples are given.     

A note on the convergence rates for empirical bayes estimators of parameters in multiple-parameter exponential families

Under suitable conditions upon prior distribution, the convergence rates for empirical Bayes estimators of parameters in multi-parameter exponential families (M-PEF) are obtained. It is shown that the assumptions Tong (1996) imposed on the marginal density can be reduced. The above result can also be extended to more general forms of M-PEF. Finally, some examples which satisfy the conditions of the theorems are given.     

Bayes estimators of the reliability of the logistic distribution

Bayes estimators of the reliability function of the logistic distribution are obtained using the methods of Lindley (1980) and Tierney & Kadane (1986). Squared-error and log-odds squared-error loss functions are used. A numerical example is presented. Comparisons are made between these two procedures, based on a Monte Carlo simulation study.     

Bayes estimators of the reliability function and parameter of inverted exponential distribution using informative and non-informative priors

In this paper, we propose Bayes estimators of the parameter and reliability function of inverted exponential distribution under the general entropy loss function for complete, type I and type II censored samples. The proposed estimators have been compared with the corresponding maximum-likelihood estimators for their simulated risks (average loss over sample space).     

Empirical Bayes estimators in hierarchical models with mixture priors

We consider subgroup analyses within the framework of hierarchical modeling and empirical Bayes (EB) methodology for general priors, thereby generalizing the normal–normal model. By doing this one obtains greater flexibility in modeling. We focus on mixture priors, that is, on the situation where group effects are exchangeable within clusters of subgroups only. We establish theoretical results on accuracy, precision, shrinkage and selection bias of EB estimators under the general priors. The impact of model misspecification is investigated and the applicability of the methodology is illustrated with datasets from the (medical) literature.